1 / 10

Testing for Unit Roots

Testing for Unit Roots. Consider an AR(1): y t = a + r y t-1 + e t Let H 0 : r = 1, (assume there is a unit root) Define q = r – 1 and subtract y t-1 from both sides to obtain D y t = a + q y t-1 + e t

Download Presentation

Testing for Unit Roots

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Testing for Unit Roots • Consider an AR(1): yt = a + ryt-1 + et • Let H0: r = 1, (assume there is a unit root) • Define q = r – 1 and subtract yt-1 from both sides to obtain Dyt = a + qyt-1 + et • Unfortunately, a simple t-test is inappropriate, since this is an I(1) process • A Dickey-Fuller Test uses the t-statistic, but different critical values Economics 20 - Prof. Anderson

  2. Testing for Unit Roots (cont) • We can add p lags of Dyt to allow for more dynamics in the process • Still want to calculate the t-statistic for q • Now it’s called an augmented Dickey-Fuller test, but still the same critical values • The lags are intended to clear up any serial correlation, if too few, test won’t be right Economics 20 - Prof. Anderson

  3. Testing for Unit Roots w/ Trends • If a series is clearly trending, then we need to adjust for that or might mistake a trend stationary series for one with a unit root • Can just add a trend to the model • Still looking at the t-statistic for q, but the critical values for the Dickey-Fuller test change Economics 20 - Prof. Anderson

  4. Spurious Regression • Consider running a simple regression of yt on xt where yt and xt are independent I(1) series • The usual OLS t-statistic will often be statistically significant, indicating a relationship where there is none • Called the spurious regression problem Economics 20 - Prof. Anderson

  5. Cointegration • Say for two I(1) processes, yt and xt, there is a b such that yt – bxt is an I(0) process • If so, we say that y and x are cointegrated, and call b the cointegration parameter • If we know b, testing for cointegration is straightforward if we define st = yt – bxt • Do Dickey-Fuller test and if we reject a unit root, then they are cointegrated Economics 20 - Prof. Anderson

  6. Cointegration (continued) • If b is unknown, then we first have to estimate b , which adds a complication • After estimating b we run a regression of Dût on ût-1 and compare t-statistic on ût-1 with the special critical values • If there are trends, need to add it to the initial regression that estimates b and use different critical values for t-statistic on ût-1 Economics 20 - Prof. Anderson

  7. Forecasting • Once we’ve run a time-series regression we can use it for forecasting into the future • Can calculate a point forecast and forecast interval in the same way we got a prediction and prediction interval with a cross-section • Rather than use in-sample criteria like adjusted R2, often want to use out-of-sample criteria to judge how good the forecast is Economics 20 - Prof. Anderson

  8. Out-of-Sample Criteria • Idea is to note use all of the data in estimating the equation, but to save some for evaluating how well the model forecasts • Let total number of observations be n + m and use n of them for estimating the model • Use the model to predict the next m observations, and calculate the difference between your prediction and the truth Economics 20 - Prof. Anderson

  9. Out-of-Sample Criteria (cont) • Call this difference the forecast error, which is ên+h+1 for h = 0, 1, …, m • Calculate the root mean square error (RMSE) Economics 20 - Prof. Anderson

  10. Out-of-Sample Criteria (cont) • Call this difference the forecast error, which is ên+h+1 for h = 0, 1, …, m • Calculate the root mean square error and see which model has the smallest, where Economics 20 - Prof. Anderson

More Related